GeRgraphy

16 March, 2016

GIS

Geographic Information Systems

GIS

Geographic Information Systems

incorporating
storing
manipulating
analyzing
displaying…

Spatial Data

What is spatial data?

Nonspatial data has no location information

nonspatial = data.frame(
  id=c(1,2,3,4),
  data=rnorm(4)
)
print(nonspatial)

##   id       data
## 1  1 -0.4081061
## 2  2 -0.8422845
## 3  3 -0.2370283
## 4  4  1.6211359

What is spatial data?

Spatial data has location information

The simplest spatial data are points on a map

spatial = data.frame(
  id=c(1,2,3,4),
  data=rnorm(4),
  x=runif(4,-180,180),
  y=runif(4,-90,90)
)
print(spatial)

##   id       data          x          y
## 1  1 -0.2084150 -172.61333 -71.594921
## 2  2 -0.5754466  -29.24202   4.005584
## 3  3 -0.1646887  -44.08515  89.482876
## 4  4 -1.4658640  -20.39301 -50.607725

What is spatial data?

Which we can convert to explicitly spatial data using the sp package. Most GIS packages in R store data as sp classes.

library(sp)

What is spatial data?

The sp package has a method called coordinates that converts points to an sp class.

coordinates(spatial) = ~ x + y
class(spatial)

## [1] "SpatialPointsDataFrame"
## attr(,"package")
## [1] "sp"

plot(spatial, axes=T)

What is spatial data?

Spatial data also needs a reference system or "projection" that allows us to represent spatial features on a map. Projections can be thought of as simply a coordinate system with an origin that is relative to a known point in space.

This is a whole field of mathematically intensive study termed "geodesy"

Much of the field of geodesy is jam-packed in the rgdal and sp packages, which use the PROJ.4 library for projections and transformations.

## will automatically load sp for us too
library(rgdal)

Wonderful document on CRS in R here

What is spatial data?

rgdal includes a comprehensive list of projections that are typically represented as a string of parameters according to the PROJ.4 specifications. The simplest is a latitude/longitude system denoted as

"+proj=longlat"

To define the projection for spatial, we write to its proj4string slot:

proj4string(spatial) = "+proj=longlat"

Projections are a necessary evil for GIS users (to be continued)

What is spatial data?

With a projection associated with our spatial data, we can now relate it to other spatial data. In other words, let's make a map!

library(leaflet)

m = leaflet(data=spatial) %>%
  addTiles() %>%
  addMarkers()
m

What is spatial data?

Arguably, the most common use of spatial data, and therefore GIS, are in data exploration, simply taking a look at where the phenomenon occur. This is useful in the sense that we can see obvious trends in the data, giving clues for further analysis, or merely checking our work, seeing if we have made mistakes. Additionally, spatial data is used for different kinds of analysis, leveraging the location data to make inferences. This analysis could take the form of selecting all wells that fall within certain soil types or in the construction of error terms in construction a statistical model.

Maybe delete this paragraph… The most common software used in GIS are ArcGIS, produced by ESRI and QGIS, an open source analog, but there are many other software packages for GIS, both propietary and open-source. Any familiarity with the most popular GISs will show you that they are GUI-based and therefore much different from the look and feel of R, and, while R is not necessarily a replacement for the ArcGIS or QGIS, the analysis and plotting capability of R makes it a powerful GIS, and when it comes to model construction, R, and no suprise here, blows the common GIS platforms out of the water.

Spatial data types

Two main types: vector and raster

Vector = Polygons
Raster = Grid

Vector = Discrete
Raster = Continuous

Vector = Illustrator/Inkscape
Raster = Photoshop/GIMP

Maybe break this apart and put this into the respective intros of the datatypes below? Spatial data are divided up into two kinds of data, vector data and raster data. Vector data consist of the location data being defined by points. There are three main kinds of vector data: points, lines and polygons. Point data, as might be guessed, are data where each observation is defined by one coordinate pair, X and Y. In line data each observation is defined by a series of coordinate pairs that define a one dimensional path in space. Polygons are data where each observation is defined by coordinate pairs that trace out a two-dimensional area. Common uses of vector data are for phenomenon that have discrete boundaries and locations, e.g., wells and bird sitings for points, rivers and streets (at certain scales) for lines, and county borders and soil types for polygon data. Raster data are constructed as grids that are laid out over an area. Each cell of the grid contains some information about the phenomenon, but each cell will not be explicitly given a coordinate pair, rather every cell has the same length and width and so only the certain corners of the data need to be given coordinate pairs. Precipitation and temperature data are often portrayed as raster data because of the way they vary continuously over the space. Examples of the creation, input, and output of these data types are given below.

Note to power users: The usual definitions of these data describe vector data as being for data with discrete boundaries and raster data for that which continuously varies. While this is true in general, it is essentially a matter of convenience that determines which type is best. For example, you could represent a dataset of bird sitings in a raster format, but because the grid is so sparse (NA or 'no siting' for nearly all the cells) it saves memory (and makes more inuitive sense) to store the data as points. Conversely, rainfall data can be stored as thousands of points, each marking the estimated rainfall at that location, but storing the data in this format is rather cumbersome if the area is large.

This presentation is meant to give an introduction to handling spatial data in R. It's intended audience are those with at least a basic knowledge of R and an interest in, though not necessarily knowledge of, working with spatial data. We will cover basic handling and manipulation of vector data and raster data, and then give a flyover of how coordinate reference systems work and how to work with them in R.

Spatial data types

Vector Data

Vector Data Intro

Whether, points, lines or polygons, comprised of coordinate pairs

Vector Data Intro

Think like a table, each row (observation) is referred to as a feature.

The other data is often referred to as the attribute table or as attribute data.

Each row is one observation (one feature), this becomes important later, as we'll need to refer to individual features or to the entire geometry.

Vector Data Basics

Data Input/Output

library(rgdal)
soils = readOGR(
    dsn="data",
    layer="soilsData")

## OGR data source with driver: ESRI Shapefile 
## Source: "data", layer: "soilsData"
## with 71 features
## It has 27 fields

writeOGR(
    soils,
    "data",
    "soilsData_out",
    driver="ESRI Shapefile"
)

Vector Data Basics

Other ways of creating of spatial data from list of coordinates:

wells = read.delim("./data/WellLocations.tsv")
class(wells); head(wells)

## [1] "data.frame"

##           x        y pts.data.id
## 1 -90.05145 43.10047           1
## 2 -90.05553 43.10470           2
## 3 -90.07305 43.09013           3
## 4 -90.04716 43.08454           4
## 5 -90.07198 43.08850           5
## 6 -90.06599 43.09197           6

coordinates(wells) <- ~ x + y
class(wells)

## [1] "SpatialPointsDataFrame"
## attr(,"package")
## [1] "sp"

Vector Data Basics

Helper functions:

class(soils)

## [1] "SpatialPolygonsDataFrame"
## attr(,"package")
## [1] "sp"

slotNames(soils)

## [1] "data"        "polygons"    "plotOrder"   "bbox"        "proj4string"

length(soils)

## [1] 71

Vector Data Basics

Accessing attribute data

str(soils@data[,1:10])

## 'data.frame':    71 obs. of  10 variables:
##  $ mukey  : Factor w/ 25 levels "2774742","2774772",..: 12 19 12 6 9 5 7 24 25 13 ...
##  $ muarcrs: Factor w/ 71 levels "0.40538405","0.90105194",..: 29 59 67 17 8 25 18 5 14 22 ...
##  $ Sand1  : num  21.5 11 21.5 12 29.5 ...
##  $ Sand2  : num  35.29 8.34 35.29 9.02 38.22 ...
##  $ Sand3  : num  45.09 7.56 45.09 16.59 64.52 ...
##  $ Sand4  : num  48.6 30.6 48.6 36.7 33.2 ...
##  $ Sand5  : num  0 31.1 0 27.1 57.2 ...
##  $ Silt1  : num  45.8 65.2 45.8 68.7 54.5 ...
##  $ Silt2  : num  37.3 64.9 37.3 60.3 43.5 ...
##  $ Silt3  : num  36.7 64.1 36.7 34 21.1 ...

Vector Data Basics

The geometry stored in the polygons slot

str(soils@polygons[1])

## List of 1
##  $ :Formal class 'Polygons' [package "sp"] with 5 slots
##   .. ..@ Polygons :List of 1
##   .. .. ..$ :Formal class 'Polygon' [package "sp"] with 5 slots
##   .. .. .. .. ..@ labpt  : num [1:2] 514199 291168
##   .. .. .. .. ..@ area   : num 10776
##   .. .. .. .. ..@ hole   : logi FALSE
##   .. .. .. .. ..@ ringDir: int 1
##   .. .. .. .. ..@ coords : num [1:21, 1:2] 514211 514206 514195 514178 514180 ...
##   .. ..@ plotOrder: int 1
##   .. ..@ labpt    : num [1:2] 514199 291168
##   .. ..@ ID       : chr "0"
##   .. ..@ area     : num 10776

Vector Data Basics

A number of common functions have methods for spatial data

silty = subset(soils, Silt1 > 70)
paste("There are", length(soils), "soil features total;")

## [1] "There are 71 soil features total;"

paste(length(silty), "with a silt percentage over 70")

## [1] "10 with a silt percentage over 70"

Vector data basics

Making simple maps is quite easy

par(mfrow=c(1,2), bg=NA)
plot(
    soils,
    main="Soils Polygons",
    col=rainbow(5))
plot(
    wells,
    main="Well Data",
    col='red'
)

Vector data basics

Coordinate Reference Systems

A (Very) Brief Break

A coordinate reference system (CRS) defines the surface of the world. If they don't match, errors and issues can arise

Coordinate Reference Systems

A (Very) Brief Break

To illustrate issues, where are the wells in relation to the soil?

soils = readOGR(
    dsn="data",
    layer="soilsData")
plot(
    soils,
    main="Soils"
)
plot(
    wells,
    add=T
)

Coordinate Reference Systems

A (Very) Brief Break

print(wells@proj4string)

## CRS arguments: NA

print(soils@proj4string)

## CRS arguments:
##  +proj=tmerc +lat_0=0 +lon_0=-90 +k=0.9996 +x_0=520000
## +y_0=-4480000 +datum=NAD83 +units=m +no_defs +ellps=GRS80
## +towgs84=0,0,0

coordinates(soils)[c(1,72)]

## [1] 514198.6 291168.3

coordinates(wells)[c(1,16)]

## [1] -90.05145  43.10047

Coordinate Reference Systems

A (Very) Brief Break

Solution: define the CRS then project the points to CRS of the soils data

wells@proj4string = CRS(
"+proj=longlat +ellps=WGS84 +datum=WGS84 +no_defs")
wells = spTransform(
    wells,
    soils@proj4string)
plot(soils)
plot(
    wells,
    add=T,
    cex=2,
    col='red',
    pch=19
)

Vector Data Geometry Relationships

How do two shapes relate? Sometimes we want to have a yes/no true/false answer to how two geometries relate. Spatial Predicates is the term for this

All contained in the rgeos package. Here we see gIntersects used on these two polygons. gEquals, gTouches, gCrosses, gContains and others.

Vector Data Example

gIntersects(soils, wells)

## [1] TRUE

head(gIntersects(wells, soils, byid=T, returnDense=T))

##       1     2     3     4     5     6     7     8     9    10    11    12
## 0 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## 1 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## 2 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## 3 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## 4 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## 5 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##      13    14    15
## 0 FALSE FALSE FALSE
## 1 FALSE FALSE FALSE
## 2 FALSE FALSE FALSE
## 3 FALSE FALSE FALSE
## 4 FALSE FALSE FALSE
## 5 FALSE FALSE FALSE

gIntersects(wells, soils, byid=T, returnDense=F)[1:2]

## $`1`
## [1] 66
## 
## $`2`
## [1] 36

Vector Data Example

gIntersects(wells, soils, byid=T, returnDense=F)[1:4]

## $`1`
## [1] 66
## 
## $`2`
## [1] 36
## 
## $`3`
## [1] 22
## 
## $`4`
## [1] 29

Vector Data Example

indxs <- gIntersects(
    
    wells,
    soils,
    byid=T,
    returnDense=F) %>%
    unlist %>%
    unique

soils_with_wells = soils[indxs,]
plot(
    soils_with_wells
)
plot(
    wells,
    add=T,
    cex=2,
    pch=19,
    col='red'
)

Vector Data Example

Extract soils data to our point data…

soilsDat = over(wells, soils)
class(soilsDat)

## [1] "data.frame"

soilsDat$id = 1:15

wells = merge(wells, soilsDat, by.x="pts.data.id", by.y="id")
head(wells@data)

##   pts.data.id   mukey      muarcrs    Sand1     Sand2    Sand3    Sand4
## 1           1 2806652 405.22965906 35.87900 32.747667 69.83400 73.27298
## 2           2 2806642 127.65139889 14.10700 14.001433 13.06847 14.28767
## 3           3 2774772  23.43535446 10.51290  9.529000 10.50538 24.74279
## 4           4 2774821   9.99030648 13.44400 11.495200  9.31480 30.69747
## 5           5 2774777   4.35597440 12.03429  9.015357 16.59286 36.73413
## 6           6 2774798   8.69344646 21.47179 35.293641 45.09015 48.59615
##      Sand5    Silt1    Silt2    Silt3    Silt4    Silt5    Clay1    Clay2
## 1 73.07354 49.00485 40.71233 15.42715 13.57983 14.06340 15.11615 26.54000
## 2 15.86000 71.64800 71.62823 70.58203 68.64867 69.06100 14.24500 14.37033
## 3 30.45305 70.65485 65.64850 65.46994 35.22879 54.23537 18.83225 24.82250
## 4 48.58000 68.94600 67.97213 66.78053 47.81453 35.96000 17.61000 20.53267
## 5 27.11190 68.71821 60.28321 34.02286 39.86052 41.12143 19.24750 30.70143
## 6  0.00000 45.78205 37.27431 36.66754 28.13462  0.00000 32.74615 27.43205
##      Clay3    Clay4    Clay5       OM1       OM2       OM3       OM4
## 1 14.73885 13.14719 12.86306 1.7208333 0.3348077 0.2575000 0.2519231
## 2 16.34950 17.06367 15.07900 2.1704167 1.9599167 2.7409167 1.8906667
## 3 24.02468 40.02842 15.31158 0.9820000 0.2550000 0.2550395 0.2535526
## 4 23.90467 21.48800 15.46000 2.7300000 1.6890000 0.5740000 0.4036667
## 5 49.38429 23.40536 31.76667 1.2584821 0.2544643 0.2500000 0.2500000
## 6 18.24231 13.01282  0.00000 0.8837179 0.2500000 0.2500000 0.2243590
##      OM5 Depth1 Depth2 Depth3 Depth4 Depth5
## 1 0.2500     39     40     40     40     40
## 2 0.2575     30     31     30     31     31
## 3 0.2500     40     40     41     40     41
## 4 0.2600     30     31     31     31     31
## 5 0.2500     28     29     28     29     29
## 6 0.0000     39     40     40     40     40

Vector Data Example

Perhaps we want to expand our our of concern

## buffer of 50 meters (because of our projection) around each well
wells_area = gBuffer(wells, byid = T, width = 50)
class(wells_area)

## [1] "SpatialPolygonsDataFrame"
## attr(,"package")
## [1] "sp"

soils@data = subset(soils@data, select = c("Sand1", "Silt1"))
head(over(wells_area, soils, fn = mean))

##      Sand1    Silt1
## 1 24.84412 58.12692
## 2 12.96008 68.01767
## 3 11.08520 69.97385
## 4 13.62663 68.09750
## 5 12.03429 68.71821
## 6 16.56465 57.53745

Vector Data

Another kind of relationship functions return the geometry (versus returning booleans) How do we get the overlapping area?

Vector Data

We use the gIntersection function, this returns that area to us xyinter = gIntersection(x, y). Other functions like gDifference, gUnion, or gSymdifference also return geometry.

Vector Data

Grab data within a certain range of the middle? Note how this returns geometry

cent = gCentroid(soils)
bff = gBuffer(cent, width = 250)
int = gIntersection(soils, bff, byid = T)
plot(int, col = rainbow(5))

Raster Data

Intro

A raster grid is rectangular.

Grid is another word for matrix.

Grid is another word for image.

A GIS raster grid is a matrix/image with an associated location and projection.

Intro

At a minimum, a GIS raster grid contains:

matrix of values
projection
reference point, often (x,y) of the lower-left corner
cellsize

Raster I/O

The rgdal rgdal packages is primarily for I/O and projecting GIS data

library(rgdal)

The raster package does everything rgdal does, but it includes lots of additional functionality.

library(raster)

Raster I/O

elev = readGDAL("data/dem_wi.tif")
writeGDAL(elev, "data/dem_wi_out.tif")

elev = raster("data/dem_wi.tif")
writeRaster(elev, "data/dem_wi_out.tif")

Raster data structure

The raster object elev has all the necessary pieces of spatial information:

elev

## class       : RasterLayer 
## dimensions  : 284, 387, 109908  (nrow, ncol, ncell)
## resolution  : 0.01666667, 0.01666667  (x, y)
## extent      : -93.03262, -86.58262, 42.3949, 47.12823  (xmin, xmax, ymin, ymax)
## coord. ref. : +proj=longlat +datum=WGS84 +no_defs +ellps=WGS84 +towgs84=0,0,0 
## data source : /home/devans/Documents/GeRgraphyPresentation/data/dem_wi.tif 
## names       : dem_wi 
## values      : 175, 565.4104  (min, max)

Raster data structure

Which means we can make a map!

m = leaflet() %>%
  addTiles() %>%
  addRasterImage(elev, opacity=0.5)
m

Raster data structure

Raster analysis

Remember that rasters are just matrices!

Therefore, most matrix operations can be applied to rasters. For example:

plot(
  elev > 400,
  col=c("red", "blue")
)

Raster analysis

Rasters can be easily converted to matrices to do more complex work.

lat_grad = apply(
  as.matrix(elev),
  1,
  mean,
  na.rm=T
)
plot(lat_grad, type="l")

Raster overlay

Most raster analysis ultimately executes some sort of overlay.

The issue:

To overlay two or more rasters, their projections, extents, and cellsizes must align perfectly.

This can be a difficult task.

Raster overlay

coordinate systems

What is the highest point in each county?

# Pseudo-code
1. Read in elevation data (raster grid)
2. Read in county boundary data (polygons)
3. Convert counties to raster grid that aligns with elevation grid
4. Find maximum elevation gridcell within each county

Raster overlay

coordinate systems

counties = readOGR("data", "WI_Counties")

## OGR data source with driver: ESRI Shapefile 
## Source: "data", layer: "WI_Counties"
## with 72 features
## It has 7 fields

elev

## class       : RasterLayer 
## dimensions  : 284, 387, 109908  (nrow, ncol, ncell)
## resolution  : 0.01666667, 0.01666667  (x, y)
## extent      : -93.03262, -86.58262, 42.3949, 47.12823  (xmin, xmax, ymin, ymax)
## coord. ref. : +proj=longlat +datum=WGS84 +no_defs +ellps=WGS84 +towgs84=0,0,0 
## data source : /home/devans/Documents/GeRgraphyPresentation/data/dem_wi.tif 
## names       : dem_wi 
## values      : 175, 565.4104  (min, max)

Raster overlay

coordinate systems

proj4string(counties)

## [1] "+proj=tmerc +lat_0=0 +lon_0=-90 +k=0.9996 +x_0=520000 +y_0=-4480000 +ellps=GRS80 +units=m +no_defs"

proj4string(elev)

## [1] "+proj=longlat +datum=WGS84 +no_defs +ellps=WGS84 +towgs84=0,0,0"

Raster overlay

coordinate systems

extent(counties)

## class       : Extent 
## xmin        : 294839 
## xmax        : 770036.4 
## ymin        : 225108.8 
## ymax        : 734398.4

extent(elev)

## class       : Extent 
## xmin        : -93.03262 
## xmax        : -86.58262 
## ymin        : 42.3949 
## ymax        : 47.12823

Raster overlay

coordinate systems

cty_grid = rasterize(counties, elev, field="COUNTY_FIP")
summary(cty_grid)

##          layer
## Min.        NA
## 1st Qu.     NA
## Median      NA
## 3rd Qu.     NA
## Max.        NA
## NA's    109908

Raster overlay

coordinate systems

prj = proj4string(elev)
cty_prj = spTransform(counties, prj)

To do this, we use the spTransform function in the sp package.

Raster overlay

coordinate systems

extent(cty_prj)

## class       : Extent 
## xmin        : -92.88924 
## xmax        : -86.8048 
## ymin        : 42.49197 
## ymax        : 47.08077

extent(elev)

## class       : Extent 
## xmin        : -93.03262 
## xmax        : -86.58262 
## ymin        : 42.3949 
## ymax        : 47.12823

Raster overlay

coordinate systems

plot(elev)
plot(cty_prj, add=TRUE)

Raster overlay

coordinate systems

cty_grid = rasterize(counties, elev, field="COUNTY_FIP")
summary(cty_grid)

##          layer
## Min.        NA
## 1st Qu.     NA
## Median      NA
## 3rd Qu.     NA
## Max.        NA
## NA's    109908

Raster overlay

coordinate systems

extent(cty_grid)

## class       : Extent 
## xmin        : -93.03262 
## xmax        : -86.58262 
## ymin        : 42.3949 
## ymax        : 47.12823

extent(elev)

## class       : Extent 
## xmin        : -93.03262 
## xmax        : -86.58262 
## ymin        : 42.3949 
## ymax        : 47.12823

Raster overlay

coordinate systems

library(dplyr)
ovly = data.frame(
  elev=getValues(elev),
  cty=getValues(cty_grid)
)

hi_pt = ovly %>%
  group_by(cty) %>%
  mutate(
    elev = (elev == max(elev, na.rm=T)) * elev
  ) %>%
  ungroup()

elev = setValues(elev, hi_pt[["elev"]])
elev[elev == 0] = NA
hi_pt_sp = rasterToPoints(elev, spatial=T)

Raster overlay

coordinate systems

Extras

Vector Data Plotting

Scenario tasked with:

Create map of percent Democratic votes by ward
Also display percent turnout

wards = readOGR("data", "WardData")

## OGR data source with driver: ESRI Shapefile 
## Source: "data", layer: "WardData"
## with 280 features
## It has 67 fields

Vector Data Plotting

options(stringsAsFactors=F)
source("./misc_scripts/function_proper_legend.r")
library(rgdal)
library(rgeos)
library(foreign)
library(classInt)
library(RColorBrewer)
library(scales)

wards@data$SEN_PERC_DEM = with(wards@data, SEN_DEM/SEN_TOT)
wards@data$SEN_PERC_TURN = with(wards@data, SEN_TOT/PERSONS1)

wards_centroids = gCentroid(wards, byid=T)
wards_centroids = SpatialPointsDataFrame(
    gCentroid(wards, byid=T), 
    over(wards_centroids, wards)
)

Vector Data Plotting

## defining number of classes
num_classes = 6
## the color palette
pal = brewer.pal(num_classes, "RdBu")
## the class intervals to use for the colors
class_ints = classIntervals(
    wards@data$SEN_PERC_DEM,
    num_classes,
    style="quantile")
## grab the colors for plotting
colrs = findColours(class_ints, pal)

Vector Data Plotting

## Custom legend formatting
source("./misc_scripts/function_proper_legend.r")
legtxt = properLegend(colrs, 2)
plot(wards,
     col=colrs,
     main="Senate 24",
     border=NA)

plot(wards_centroids,
     pch=20,
     cex=(wards_centroids@data$SEN_PERC_TURN),
     col=alpha('black', 0.5),
     add=T)

legend("topleft",
       legtxt,
       title="Proportion Democrat",
       fill=pal,
       bty='n'
)
legend("topright",
       c("20%", '50%', '80%'),
       pch=20,
       pt.cex=c(0.2, 0.5, 0.8),
       col=alpha('black', 0.5),
       bty='n',
       title="Percent\nTurnout"
)

Vector Data Plotting

Vector Data Modeling

library(spdep)

## Loading required package: Matrix

wards@data$SEN_PERC_DEM = with(wards@data, SEN_DEM/SEN_TOT)
wards@data$SEN_PERC_TURN = with(wards@data, SEN_TOT/PERSONS1)
wards@data$CON_PERC_DEM = with(wards@data, CON_DEM/CON_TOT)
wards@data$PRES_PERC_DEM = with(wards@data, PRES_DE/PRES_TO)

### Remove NaN's created from x/0
wards = subset(wards, !is.nan(wards@data$SEN_PERC_DEM))
### Create the neighborhood structure,
###     which defines the error structure
###     for the spatial regression
neighborhood_binary = poly2nb(wards)
list_of_weights = nb2listw(neighborhood_binary, zero.policy=T)
### Run Moran's i to see if there is a spatial component to % dem
moran.test(wards@data$SEN_PERC_DEM, list_of_weights, alternative="two.sided")

## 
##  Moran I test under randomisation
## 
## data:  wards@data$SEN_PERC_DEM  
## weights: list_of_weights  
## 
## Moran I statistic standard deviate = 15.295, p-value < 2.2e-16
## alternative hypothesis: two.sided
## sample estimates:
## Moran I statistic       Expectation          Variance 
##       0.567975167      -0.003623188       0.001396579

### Run spatial regression
spat_lin_reg = spautolm(
    SEN_PERC_DEM ~ WHITE + BLACK + PRES_PERC_DEM + CON_PERC_DEM,
    data=wards,
    family="SAR",
    listw=list_of_weights)